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May, 1973 On Consistency in Monotonic Regression
D. L. Hanson, Gordon Pledger, F. T. Wright
Ann. Statist. 1(3): 401-421 (May, 1973). DOI: 10.1214/aos/1176342407


For each $t$ in some subset $T$ of $N$-dimensional Euclidean space let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is non-decreasing in each of the coordinates of $t$. Let $t_1, t_2,\cdots$ be a sequence of points in $T$ and let $Y_1, Y_2,\cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. Estimators $\hat{m}_n(t; Y_1,\cdots, Y_n)$ of $m(t)$ which are monotone in each coordinate of $t$ and which minimize $\sum^n_{i=1} \lbrack\hat{m}_n(t_i; Y_1,\cdots, Y_n) - Y_i\rbrack^2$ are already known. Brunk has investigated their consistency when $N = 1$. In this paper additional consistency results are obtained when $N = 1$ and some results are obtained in the case $N = 2$. In addition, we prove several lemmas about the law of large numbers which we believe to be of independent interest.


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D. L. Hanson. Gordon Pledger. F. T. Wright. "On Consistency in Monotonic Regression." Ann. Statist. 1 (3) 401 - 421, May, 1973.


Published: May, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0259.62037
MathSciNet: MR353540
Digital Object Identifier: 10.1214/aos/1176342407

Primary: 62G05
Secondary: 60G50

Keywords: consistency , isotonic regression , Law of Large Numbers , ‎mean‎ , Monotonic regression

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 3 • May, 1973
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